Generating area-preserving maps of the plane

Question

What's the most efficient way of generating area-preserving maps of the plane? I know that Hamiltonian flows are area-preserving, but are all smooth, well-behaved, orientation-preserving area-preserving maps Hamiltonian flows? I know that if I want an angle-preserving map I can just choose a holomorphic function of the form $f(z) = \sum_n c_n z^n$, and in that way I can quickly generate many angle-preserving maps. Is there anything analogous for area-preserving maps?